Coronal radiative losses
Encyclopedia
In astronomy
and in astrophysics
, for radiative losses of the solar corona, it is meant the energy flux
irradiated from the external atmosphere
of the Sun
(traditionally divided into chromosphere
, transition region and corona
), and, in particular, the processes of production of the radiation
coming from the solar corona and transition region, where the plasma is optically-thin. On the contrary, in the chromosphere, where the temperature decreases from the photospheric value of 6000 K to the minimum of 4400 K, the optical depth
is about 1, and the radiation is thermal.
The corona
extends much further than a solar radius from the photosphere
and looks very complex and inhomogeneous in the X-rays images taken by satellites (see the figure on the right taken by the XRT on board Hinode
).
The structure and dynamics of the corona
are dominated by the solar magnetic field. There are strong evidences that even the heating mechanism, responsible for its high temperature of million degrees, is linked to the magnetic field of the Sun
.
The energy flux
irradiated from the corona changes in active regions, in the quiet Sun and in coronal holes
; actually, part of the energy is irradiated outwards, but approximatively the same amount of the energy flux is conducted back towards the chromosphere
, through the steep transition region. In active regions the energy flux is about 107 erg cm−2sec−1, in the quiet Sun it is roughly 8 105 - 106 erg cm−2sec−1, and in coronal holes 5 105 - 8 105 erg cm−2sec−1, including the losses due to the solar wind.
The required power is a small fraction of the total flux irradiated from the Sun, but this energy is enough to maintain the plasma at the temperature of million degrees, since the density is very low and the processes of radiation are different from those occurring in the photosphere, as it is shown in detail in the next section.
are emitted mainly in the X-rays. This radiation is not visible from the Earth
because it is filtered by the atmosphere
. Before the first rocket missions, the corona could be observed only in white light during the eclipses, while in the last fifty years the solar corona has been photographed in the EUV and X-rays by many satellites (Pioneer
5, 6, 7, 8, 9, Helios
, Skylab
, SMM
, NIXT
, Yohkoh
, SOHO
, TRACE
, Hinode
).
The emitting plasma is almost completely ionized and very light, its density is about 10−16 - 10−14 g/cm3. Particles are so isolated that almost all the photons can leave the Sun
's surface without interacting with the matter above the photosphere
: in other words, the corona is transparent to the radiation and the emission of the plasma
is optically-thin. The Sun's atmosphere is not the unique example of X-ray source, since hot plasmas are present wherever in the Universe: from stellar coronae to thin galactic halo
s. These stellar environments are the subject of the X-ray astronomy
.
In an optically-thin plasma the matter is not in thermodynamical equilibrium with the radiation, because collisions between particles and photons are very rare, and, as a matter of fact, the square root mean velocity of photons, electrons, protons and ions is not the same: we should define a temperature for each of these particle populations. The result is that the emission spectrum
does not fit the spectral distribution of a blackbody radiation, but it depends only on those collisional processes which occur in a very rarefied plasma.
While the Fraunhofer lines
coming from the photosphere
are absorption lines, principally emitted from ions which absorb photons of the same frequency of the transition to an upper energy level, coronal lines are emission lines produced by metal ions which had been excited to a superior state by collisional processes. Many spectral lines are emitted by highly ionized atoms, like calcium and iron, which have lost most of their external electrons; these emission lines can be formed only at certain temperatures, and therefore their individuation in solar spectra
is sufficient to determine the temperature of the emitting plasma.
Some of these spectral lines can be forbidden on the Earth: in fact, collisions between particles can excite ions to metastable states; in a dense gas these ions immediately collide with other particles and so they de-excite with an allowed transition to an intermediate level, while in the corona it is more probable that this ion remains in its metastable state, until it encounters a photon of the same frequency of the forbidden transition to the lower state. This photon induces the ion to emit with the same frequency by stimulated emission
. Forbidden transitions from metastable states are often called as satellite lines.
The Spectroscopy
of the corona allows the determination of many physical parameters of the emitting plasma. Comparing the intensity
in lines of different ions of the same element, temperature and density can be measured with a good approximation: the different states of ionization are regulated by the Saha equation.
The Doppler shift gives a good measurement of the velocities along the line of sight
but not in the perpendicular plane.
The line width should depend on the Maxwell-Boltzmann distribution of velocities at the temperature of line formation (thermal line broadening), while it is often larger than predicted.
The widening can be due to pressure broadening, when collisions between particles are frequent, or it can be due to turbulence
: in this case the line width can be used to estimate the macroscopic velocity also on the Sun's surface, but with a great uncertainty.
The magnetic field can be measured thanks to the line splitting due to the Zeeman effect
.
are
Therefore, the radiative flux can be expressed as the sum of three terms:
where
is the number of electrons per unit volume, 
the ion
number density,
the Planck constant
,
the frequency of the emitted radiation corresponding to the energy jump 
, 
the coefficient of collisional de-excitation relative to the ion transition, 
the radiative losses for plasma recombination
and
the bremstrahlung contribution.
The first term is due to the emission in every single spectral line
. With a good approximation, the number of occupied states at the superior level
and the number of states at the inferior energy level 
are given by the equilibrium between collisional excitation and spontaneous emission
where
is the transition probability of spontaneous emission.
The second term
is calculated as the energy emitted per unit volume and time when free electrons are captured from ions to recombinate into neutral atoms (dielectronic capture).
The third term
is due to the electron scattering by protons and ions because of the Coulomb force: every accelerated charge emits radiation according to classical elettrodynamics. This effect gives an appreciable contribution to the continuum spectrum only at the highest temperatures, above 10 MK.
Taking into account all the dominant radiation processes, including satellite lines from metastable states, the emission of an optically-thin plasma can be expressed more simply as
where
depends only on the temperature. In fact, all the radiation mechanisms require collisional processes and basically depend on the squared density (
). The integral of the squared density along the line of sight is called emission measure and is often used in X-ray astronomy
.
The function
has been modeled by many authors but many discrepancies are still in these calculations: differences derive essentially on the spectral lines they include in their models and on the atomic parameters they use.
In order to calculate the radiative flux from an optically-thin plasma, it can be used the linear fitting applied to some model calculations by Rosner et al. (1978)
.
In c.g.s. unit, in erg cm3 s−1, the function P(T) can be approximated as:
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
and in astrophysics
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...
, for radiative losses of the solar corona, it is meant the energy flux
Energy flux
Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:# Rate of energy transfer per unit area...
irradiated from the external atmosphere
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
(traditionally divided into chromosphere
Chromosphere
The chromosphere is a thin layer of the Sun's atmosphere just above the photosphere, roughly 2,000 kilometers deep....
, transition region and corona
Corona
A corona is a type of plasma "atmosphere" of the Sun or other celestial body, extending millions of kilometers into space, most easily seen during a total solar eclipse, but also observable in a coronagraph...
), and, in particular, the processes of production of the radiation
Radiation
In physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...
coming from the solar corona and transition region, where the plasma is optically-thin. On the contrary, in the chromosphere, where the temperature decreases from the photospheric value of 6000 K to the minimum of 4400 K, the optical depth
Optical depth
Optical depth, or optical thickness, is a measure of transparency. Optical depth is defined by the negative logarithm of the fraction of radiation that is not scattered or absorbed on a path...
is about 1, and the radiation is thermal.
The corona
Corona
A corona is a type of plasma "atmosphere" of the Sun or other celestial body, extending millions of kilometers into space, most easily seen during a total solar eclipse, but also observable in a coronagraph...
extends much further than a solar radius from the photosphere
Photosphere
The photosphere of an astronomical object is the region from which externally received light originates. The term itself is derived from Ancient Greek roots, φῶς, φωτός/phos, photos meaning "light" and σφαῖρα/sphaira meaning "sphere", in reference to the fact that it is a spheric surface perceived...
and looks very complex and inhomogeneous in the X-rays images taken by satellites (see the figure on the right taken by the XRT on board Hinode
Hinode
Hinode , formerly Solar-B, is a Japan Aerospace Exploration Agency Solar mission with United States and United Kingdom collaboration. It is the follow-up to the Yohkoh mission and it was launched on the final flight of the M-V-7 rocket from Uchinoura Space Center, Japan on 22 September 2006 at...
).
The structure and dynamics of the corona
Corona
A corona is a type of plasma "atmosphere" of the Sun or other celestial body, extending millions of kilometers into space, most easily seen during a total solar eclipse, but also observable in a coronagraph...
are dominated by the solar magnetic field. There are strong evidences that even the heating mechanism, responsible for its high temperature of million degrees, is linked to the magnetic field of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
.
The energy flux
Energy flux
Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:# Rate of energy transfer per unit area...
irradiated from the corona changes in active regions, in the quiet Sun and in coronal holes
Coronal holes
Coronal holes are areas where the Sun's corona is darker, colder, and has lower-density plasma than average. These were found when X-ray telescopes in the Skylab mission were flown above the Earth's atmosphere to reveal the structure of the corona. Coronal holes are linked to unipolar...
; actually, part of the energy is irradiated outwards, but approximatively the same amount of the energy flux is conducted back towards the chromosphere
Chromosphere
The chromosphere is a thin layer of the Sun's atmosphere just above the photosphere, roughly 2,000 kilometers deep....
, through the steep transition region. In active regions the energy flux is about 107 erg cm−2sec−1, in the quiet Sun it is roughly 8 105 - 106 erg cm−2sec−1, and in coronal holes 5 105 - 8 105 erg cm−2sec−1, including the losses due to the solar wind.
The required power is a small fraction of the total flux irradiated from the Sun, but this energy is enough to maintain the plasma at the temperature of million degrees, since the density is very low and the processes of radiation are different from those occurring in the photosphere, as it is shown in detail in the next section.
Processes of radiation of the solar corona
The electromagnetic waves coming from the solar coronaCorona
A corona is a type of plasma "atmosphere" of the Sun or other celestial body, extending millions of kilometers into space, most easily seen during a total solar eclipse, but also observable in a coronagraph...
are emitted mainly in the X-rays. This radiation is not visible from the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
because it is filtered by the atmosphere
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
. Before the first rocket missions, the corona could be observed only in white light during the eclipses, while in the last fifty years the solar corona has been photographed in the EUV and X-rays by many satellites (Pioneer
Pioneer program
The Pioneer program is a series of United States unmanned space missions that was designed for planetary exploration. There were a number of such missions in the program, but the most notable were Pioneer 10 and Pioneer 11, which explored the outer planets and left the solar system...
5, 6, 7, 8, 9, Helios
Helios
Helios was the personification of the Sun in Greek mythology. Homer often calls him simply Titan or Hyperion, while Hesiod and the Homeric Hymn separate him as a son of the Titans Hyperion and Theia or Euryphaessa and brother of the goddesses Selene, the moon, and Eos, the dawn...
, Skylab
Skylab
Skylab was a space station launched and operated by NASA, the space agency of the United States. Skylab orbited the Earth from 1973 to 1979, and included a workshop, a solar observatory, and other systems. It was launched unmanned by a modified Saturn V rocket, with a mass of...
, SMM
Solar Maximum Mission
The Solar Maximum Mission satellite was designed to investigate solar phenomenon, particularly solar flares. It was launched on February 14, 1980....
, NIXT
NIXT
The NIXT, or Normal Incidence X-ray Telescope, was a sounding rocket payload flown in the 1990s by Professor Leon Golub of the Smithsonian Astrophysical Observatory, to prototype normal-incidence optical designs in extreme ultraviolet solar imaging...
, Yohkoh
Yohkoh
Yohkoh , known before launch as Solar-A, was a Solar observatory spacecraft of the Institute of Space and Astronautical Science with United States and United Kingdom collaboration...
, SOHO
Solar and Heliospheric Observatory
The Solar and Heliospheric Observatory is a spacecraft built by a European industrial consortium led by Matra Marconi Space that was launched on a Lockheed Martin Atlas IIAS launch vehicle on December 2, 1995 to study the Sun, and has discovered over 2100 comets. It began normal operations in May...
, TRACE
TRACE
TRACE was a NASA space telescope designed to investigate the connections between fine-scale magnetic fields and the associated plasma structures on the Sun by providing high resolution images and observation of the solar photosphere and transition region to the corona...
, Hinode
Hinode
Hinode , formerly Solar-B, is a Japan Aerospace Exploration Agency Solar mission with United States and United Kingdom collaboration. It is the follow-up to the Yohkoh mission and it was launched on the final flight of the M-V-7 rocket from Uchinoura Space Center, Japan on 22 September 2006 at...
).
The emitting plasma is almost completely ionized and very light, its density is about 10−16 - 10−14 g/cm3. Particles are so isolated that almost all the photons can leave the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
's surface without interacting with the matter above the photosphere
Photosphere
The photosphere of an astronomical object is the region from which externally received light originates. The term itself is derived from Ancient Greek roots, φῶς, φωτός/phos, photos meaning "light" and σφαῖρα/sphaira meaning "sphere", in reference to the fact that it is a spheric surface perceived...
: in other words, the corona is transparent to the radiation and the emission of the plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
is optically-thin. The Sun's atmosphere is not the unique example of X-ray source, since hot plasmas are present wherever in the Universe: from stellar coronae to thin galactic halo
Galactic halo
The term galactic halo is used to denote an extended, roughly spherical component of a galaxy, which extends beyond the main, visible component. It can refer to any of several distinct components which share these properties:* the galactic spheroid...
s. These stellar environments are the subject of the X-ray astronomy
X-ray astronomy
X-ray astronomy is an observational branch of astronomy which deals with the study of X-ray observation and detection from astronomical objects. X-radiation is absorbed by the Earth's atmosphere, so instruments to detect X-rays must be taken to high altitude by balloons, sounding rockets, and...
.
In an optically-thin plasma the matter is not in thermodynamical equilibrium with the radiation, because collisions between particles and photons are very rare, and, as a matter of fact, the square root mean velocity of photons, electrons, protons and ions is not the same: we should define a temperature for each of these particle populations. The result is that the emission spectrum
Emission spectrum
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted by the element's atoms or the compound's molecules when they are returned to a lower energy state....
does not fit the spectral distribution of a blackbody radiation, but it depends only on those collisional processes which occur in a very rarefied plasma.
While the Fraunhofer lines
Fraunhofer lines
In physics and optics, the Fraunhofer lines are a set of spectral lines named for the German physicist Joseph von Fraunhofer . The lines were originally observed as dark features in the optical spectrum of the Sun....
coming from the photosphere
Photosphere
The photosphere of an astronomical object is the region from which externally received light originates. The term itself is derived from Ancient Greek roots, φῶς, φωτός/phos, photos meaning "light" and σφαῖρα/sphaira meaning "sphere", in reference to the fact that it is a spheric surface perceived...
are absorption lines, principally emitted from ions which absorb photons of the same frequency of the transition to an upper energy level, coronal lines are emission lines produced by metal ions which had been excited to a superior state by collisional processes. Many spectral lines are emitted by highly ionized atoms, like calcium and iron, which have lost most of their external electrons; these emission lines can be formed only at certain temperatures, and therefore their individuation in solar spectra
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
is sufficient to determine the temperature of the emitting plasma.
Some of these spectral lines can be forbidden on the Earth: in fact, collisions between particles can excite ions to metastable states; in a dense gas these ions immediately collide with other particles and so they de-excite with an allowed transition to an intermediate level, while in the corona it is more probable that this ion remains in its metastable state, until it encounters a photon of the same frequency of the forbidden transition to the lower state. This photon induces the ion to emit with the same frequency by stimulated emission
Stimulated emission
In optics, stimulated emission is the process by which an atomic electron interacting with an electromagnetic wave of a certain frequency may drop to a lower energy level, transferring its energy to that field. A photon created in this manner has the same phase, frequency, polarization, and...
. Forbidden transitions from metastable states are often called as satellite lines.
The Spectroscopy
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
of the corona allows the determination of many physical parameters of the emitting plasma. Comparing the intensity
Luminous intensity
In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye...
in lines of different ions of the same element, temperature and density can be measured with a good approximation: the different states of ionization are regulated by the Saha equation.
The Doppler shift gives a good measurement of the velocities along the line of sight
Line-of-sight propagation
Line-of-sight propagation refers to electro-magnetic radiation or acoustic wave propagation. Electromagnetic transmission includes light emissions traveling in a straight line...
but not in the perpendicular plane.
The line width should depend on the Maxwell-Boltzmann distribution of velocities at the temperature of line formation (thermal line broadening), while it is often larger than predicted.
The widening can be due to pressure broadening, when collisions between particles are frequent, or it can be due to turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
: in this case the line width can be used to estimate the macroscopic velocity also on the Sun's surface, but with a great uncertainty.
The magnetic field can be measured thanks to the line splitting due to the Zeeman effect
Zeeman effect
The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field...
.
Optically-thin plasma emission
The most important processes of radiation for an optically-thin plasmaare
- the emission in resonance lines of ionized metals (bound-bound emission);
- the radiative recombinations (free-bound radiation) due to the most abundant coronal ions;
- for very high temperatures above 10 MK, the bremstrahlung (free-free emission).
Therefore, the radiative flux can be expressed as the sum of three terms:
where
is the number of electrons per unit volume, the ionIon
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...
number density,
the Planck constantPlanck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
,
the frequency of the emitted radiation corresponding to the energy jump , the coefficient of collisional de-excitation relative to the ion transition, the radiative losses for plasma recombinationPlasma recombination
Plasma recombination is a process by which ions of a plasma capture the free energetic electrons to form new neutral atoms.Recombination usually take place in the whole volume of a plasma , although in some cases it is confined to some special region of it...
and
the bremstrahlung contribution.The first term is due to the emission in every single spectral line
Spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from a deficiency or excess of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
. With a good approximation, the number of occupied states at the superior level
and the number of states at the inferior energy level are given by the equilibrium between collisional excitation and spontaneous emissionSpontaneous emission
Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...
where
is the transition probability of spontaneous emission.
The second term
is calculated as the energy emitted per unit volume and time when free electrons are captured from ions to recombinate into neutral atoms (dielectronic capture).The third term
is due to the electron scattering by protons and ions because of the Coulomb force: every accelerated charge emits radiation according to classical elettrodynamics. This effect gives an appreciable contribution to the continuum spectrum only at the highest temperatures, above 10 MK.Taking into account all the dominant radiation processes, including satellite lines from metastable states, the emission of an optically-thin plasma can be expressed more simply as
where
depends only on the temperature. In fact, all the radiation mechanisms require collisional processes and basically depend on the squared density (). The integral of the squared density along the line of sight is called emission measure and is often used in X-ray astronomyX-ray astronomy
X-ray astronomy is an observational branch of astronomy which deals with the study of X-ray observation and detection from astronomical objects. X-radiation is absorbed by the Earth's atmosphere, so instruments to detect X-rays must be taken to high altitude by balloons, sounding rockets, and...
.
The function
has been modeled by many authors but many discrepancies are still in these calculations: differences derive essentially on the spectral lines they include in their models and on the atomic parameters they use.In order to calculate the radiative flux from an optically-thin plasma, it can be used the linear fitting applied to some model calculations by Rosner et al. (1978)
.
In c.g.s. unit, in erg cm3 s−1, the function P(T) can be approximated as:
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